Multiscale asymptotic solutions to the Boltzmann equation for - - PowerPoint PPT Presentation

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Multiscale asymptotic solutions to the Boltzmann equation for aerospace applications

Thierry MAGIN

Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics, Belgium

Issues in Solving the Boltzmann Equation for Aerospace Applications

ICERM Topical Workshop Brown University, Providence, June 3-7, 2013

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 1 / 59

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Acknowledgement

Acknowledgement

THANK YOU! Workshop organizers for this invitation to ICERM Collaborators who contributed to the results presented here

Alessandro Munaf`

• , Erik Torres and JB Scoggins (VKI)

Benjamin Graille (Paris-Sud Orsay) Marc Massot (Ecole Centrale Paris) Vincent Giovangigli (Ecole Polytechnique) Irene Gamba and Jeff Haack (The University of Texas at Austin) Marco Panesi (University of Illinois at Urbana-Champaign) Rich Jaffe, David Schwenke, Winifred Huo (NASA ARC)

Support from the European Research Council through Starting Grant #259354

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Acknowledgement

Outline

1 Introduction 2 Translational thermal nonequilibrium in plasmas 3 Atomic ionization reactions 4 Internal energy excitation in molecular gases 5 Conclusion

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Introduction Motivation

Motivation: new challenges for aerospace science

Design of spacecraft heat shields

Modeling of the convective and radiative heat fluxes for:

Robotic missions aiming at bringing back samples to Earth Manned exploration program to the Moon and Mars

Intermediate eXperimental Vehicle of ESA NASA Mars Science Laboratory

Hypersonic cruise vehicles

Modeling of flows from continuum to rarefied conditions for the next generation of air breathing hypersonic vehicles

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 4 / 59

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Introduction Motivation

Engineering design in hypersonics

Blast capsule flow simulation

VKI COOLFluiD platform and Mutation library

Two quantities of interest relevant to rocket scientists

Heat flux Shear stress to the vehicle surface

⇒ Complex multiscale problem

Chemical nonequilibrium (gas)

Dissociation, ionization, . . . Internal energy excitation

Thermal nonequilibrium

Translational and internal energy relaxation

Radiation Gas / surface interaction

Surface catalysis Ablation

Rarefied gas effects Turbulence (transition)

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Introduction Objective

Physico-chemical models for atmospheric entry plasmas

Earth atmosphere: S = {N2, O2, NO, N, O, NO+, N+, O+, e−, . . .} Fluid dynamics ρi(x, t), i ∈ S, v(x, t), E(x, t) Kinetic theory fi(x, ci, t) , i ∈ S

Fluid dynamical description

Gas modeled as a continuum in terms of macroscopic variables e.g. Navier-Stokes eqs., Boltzmann moment systems

Kinetic description

Gas particles of species i ∈ S follow a velocity distribution fi in the phase space (x, ci) e.g. Boltzmann eq.

⇒ Constraint: descriptions with consistent physico-chemical models

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Introduction Objective

From microscopic to macroscopic quantities

Velocity distribution function for 1D Ar shockwave (Mach 3.38) at different positions x ∈ [−1cm, +1cm] [Munafo et al. 2013] Mass density of species i ∈ S: ρi(x, t) = R f

i mi dci

Mixture mass density: ρ(x, t) = P

j∈S ρj(x, t)

Hydrodynamic velocity: ρ(x, t)v(x, t) = P

j∈S

R f

j mjcj dcj

Total energy (point particles): E(x, t) = P

j∈S

R f

j 1 2 mj|cj|2 dcj

Thermal (translational) energy: ρ(x, t)e(x, t) = P

j∈S

R f

j 1 2 mj|cj − v|2 dcj

⇒ Suitable asymptotic solutions can be derived by means of the Chapman-Enskog perturbative solution method

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Introduction Objective

Objective of this presentation

“Engineers use knowledge primarily to design, produce, and operate artifacts. . . Scientists, by contrast, use knowledge primarily to generate more knowledge.” Walter Vincenti

Derive asymptotic (hydrodynamic) solutions for atmospheric entry plasma flows from multiscale kinetic equations with entangled collision operators

1

Translational thermal nonequilibrium in plasmas and electro-magnetic field influence

2

Atomic ionization reactions

3

Internal energy excitation in molecular gases

⇒ Enrich mathematical models by adding more physics ⇒ Derive mathematical structure and fix ad-hoc terms found in engineering models ⇒ Integrate quantum chemistry databases

“The numerical integration of the Boltzmann equation is a matter of national prestige. . .” attributed to Anatoly Dorodnitsyn

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Translational thermal nonequilibrium in plasmas

• 1. Translational thermal nonequilibrium in plasmas

and electro-magnetic field influence

[Magin, Graille, Massot, AIAA 2008] [Magin, Graille, Massot, NASA/TM-214578 2008] [Graille, Magin, Massot, M3AS 2009] Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 9 / 59

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Translational thermal nonequilibrium in plasmas Assumptions

Boltzmann equation

Grad-Boltzmann limit: huge number of particles of very small diameter interacting with a finite mean free path value

N0 particles of characteristic diameter d0 in a box of size L0 Ratio of the olume occupied by these particles and the volume of the box:

1 6 πN0(d0/L0)3

Limits N0 → ∞ and d0/L0 → 0 for finite 1/Kn = N0π(d0/L0)2 This volume ratio can be expressed as 1

6 (d0/L0)[N0π(d0/L0)2] = 1 6 (d0/L0)/Kn

This quantity tends to zero for a fixed value of the Kn number

⇒ the gas is dilute

Assumptions

Dilute gas of n

S species: e.g. i ∈ S = {N2, O2, NO, N, O, N+ 2 , O+ 2 , NO+, N+, O+, e−}

Point particles of mass mi: no internal energy Binary and elastic collisions (no chemical reactions) Molecular chaos: probabilities of finding particle i at point (x, ci) and particle j at point (x, cj) in the phase space are independent Collision pairs of arbitrary impact parameter are equiprobable: quantity f

i (x, ci) does not

change over distances of the order of the collision cross section

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Translational thermal nonequilibrium in plasmas Assumptions

Translational thermal nonequilibrium and electromagnetic field influence in multicomponent plasma flows

Plasma composed of electrons (index e), and heavy particles, atoms and molecules, neutral or ionized (set of indices H); the full mixture

• f species is denoted by the set S = {e} ∪ H

Scaling parameter: ε = (m0

e/m0 h)1/2 ≪ 1

1

Classical mechanics description provided that

1 (n0)1/3 ≫ (m0

hkBT0)1/2

hP

and

kBT 0 m0

e

≪ c2

2

Binary charged interactions with screening of the Coulomb potential

Λ ≃ n0

e 4 3πλ3 Debye ≫ 1 3

Reference electrical and thermal energies of the system are of the same order

q0E0L0 ≃ kBT0

4

Magnetic field influence determined by the Hall parameter magnitude b

βe = q0B0

m0

e t0

e = ε1−b

(b < 0, b = 0, b = 1)

5

Continuum description for compressible flows: O(Mh) ≫ ε

Kn Mh ≃ ε

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Translational thermal nonequilibrium in plasmas Dimensional analysis

Dimensional analysis of the Boltzmann eq. [Petit, Darrozes 1975]

2 thermal speeds V 0

e =

• kBT 0

m0

e

, V 0

h =

• kBT 0

m0

h

= εV 0

e ,

ε =

• m0

e

m0

h

2 kinetic temporal scales t0

e = l0

V 0

e

, t0

h = l0

V 0

h

= t0

e

ε with l0 = 1 n0σ0 1 macroscopic temporal scale t0 = L0 v0 = L0 l0 l0 V 0

h

V 0

h

v0 = 1 Knt0

h

1 Mh = t0

h

ε

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Translational thermal nonequilibrium in plasmas Dimensional analysis

Change of variable: heavy-particle velocity frame [M3AS 2009]

The peculiar velocities are given by the relations C

e = ce − εMhvh,

C

i = ci − Mhvh,

i ∈ H ⇒ The heavy-particle diffusion flux vanishes

• j∈H
• mjC

jf j dC j = 0

The choice of the heavy-particle velocity frame vh is natural for

• plasmas. In this frame:

Heavy particles thermalize All particles diffuse

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Translational thermal nonequilibrium in plasmas Dimensional analysis

Boltzmann equation: nondimensional form and scaling

Electrons: e ∂tf

e + 1 εMh (C e + εMhvh)·∂xf e +

ε−b MhKnqe

• (C

e + εMhvh)∧B

• ·∂C

ef

e

+

• 1

εMh qeE − εMh Dvh Dt

• ·∂C

ef

e − (∂C

ef

e ⊗C e):∂xvh = 1 εMhKnJe

Heavy particles: i ∈ H ∂tf

i + 1 Mh (C i + Mhvh)·∂xf i + ε2−b

MhKn

qi mi

• (C

i + Mhvh)∧B

• ·∂C

if

i

+ 1

Mh qi mi E − Mh Dvh Dt

• ·∂C

if

i − (∂C

if

i ⊗C i):∂xvh = 1 MhKnJi

⇒ The multiscale analysis (ε, Kn, βe) occurs at three levels

in the kinetic eqs. in the crossed collision operators in the collisional invariants

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Translational thermal nonequilibrium in plasmas Dimensional analysis

Boltzmann equation: nondimensional form and scaling

Collision operators: Je = Jee (f

e, f e) + j∈H

Jej (f

e, f j )

Ji = 1

εJie(f i , f e) + j∈H

Jij(f

i , f j ),

i ∈ H Jee and Jij, i, j ∈ H, are dealt with as usual Jei and Jie, i ∈ H, depend on ε

Theorem (Degond, Lucquin 1996, Graille, M., Massot 2009)

The crossed collision operators can be expanded in the form: Jei(f

e, f i )

= J0

ei(f e, f i )(ce) + εJ1 ei(f e, f i )(ce) + ε2J2 ei(f e, f i )(ce)

+ε3J3

ei(f e, f i )(C e) + O(ε4)

Jie(f

i , f e)

= εJ1

ie(f i , f e)(ci) + ε2J2 ie(f i , f e)(ci) + ε3J3 ie(f i , f e)(C i) + O(ε4)

where i ∈ H

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Translational thermal nonequilibrium in plasmas Multiscale expansion

Enskog expansion [Graille, M., Massot 2009]

Enskog expansion (Kn Mh = ε) f

e

= f 0

e (1 + εφe + ε2φ(2) e ) + O(ε3)

fi = f 0

i (1 + εφi) + O(ε2),

i ∈ H Boltzmann eq. (βe = ε1−b)

Electron e ε−2D−2

e (f 0 e ) + ε−1D−1 e (f 0 e , φe) + D0 e(f 0 e , φe, φ(2) e )

+εD1

e(f 0 e , φe, φ(2) e , φ(3) e ) = ε−2J−2 e

+ ε−1J−1

e

+ J0

e + εJ1 e + O(ε2)

Heavy particles i ∈ H D0

i (f 0 i ) + εD1 i (f 0 i , φi) = ε−1J(−1) i

+ J0

i + εJ1 i + O(ε2)

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Translational thermal nonequilibrium in plasmas Multiscale expansion

Generalized Chapman-Enskog method [Graille, M., Massot 2009]

Order Time Heavy particles Electrons ε−2 te –

• Eq. for f 0

e

Thermalization (Te) ε−1 t0

h

• Eq. for f 0

i , i ∈ H

• Eq. for φe

Thermalization (Th) Electron momentum relation ε0 t0

• Eq. for φi, i ∈ H
• Eq. for φ(2)

e

Euler eqs. Zero-order drift-diffusion eqs. ε

t0 ε

Navier-Stokes eqs. 1st-order drift-diffusion eqs.

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Translational thermal nonequilibrium in plasmas Thermalization

Thermalization

Two quasi-equilibrium states are derived: Maxwell-Boltzmann distributions at distinct temperatures Te and Th At order ε−2, the electrons thermalize in any reference frame f 0

e = ne

• 1

2πTe 3/2 exp

• − 1

2Te C

e·C e

• The entropy production due to collisions with heavy particle vanishes

separately for each species i ∈ H At order ε−1, the heavy particles thermalize in the vh reference frame f 0

i = ni

mi 2πTh 3/2 exp

• − mi

2Th C

i·C i

• ,

i ∈ H

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Translational thermal nonequilibrium in plasmas Thermalization

Collisional invariants

Electron and heavy-particle linearized collision operators

Fe(φe) = − Z f 0

e1

• φ′

e + φ′ e1 − φe − φ e1

• |C

e − C e1|σee1 dωdC e1

− X

j∈H

nj Z σej

• |C

e|2, ω· C

e

|C

e|

” |C

e|

` φe(|C

e|ω) − φe(C e)

´ dω Fh(φ) = −[ X

j∈H

Z f 0

j

“ φ′

i + φ′ j − φi − φj

” |C

i − C j|σijdωdC j]i∈H

Collisional invariants

ˆ ψ1

e

= 1 ˆ ψ2

e

=

1 2 C e·C e

ˆ ψl

h

=

• miδil
• i∈H,

l∈H ˆ ψn

H+ν h

=

• miC

iν

• i∈H,

ν∈{1,2,3} ˆ ψn

H+4 h

=

• 1

2 miC i·C i

• i∈H

Properties

• Fe(φe), ˆ

ψl

e

e = 0, l ∈ {1, 2}

• Fh(φh), ˆ

ψl

h

h = 0, l ∈ {1, . . . , n

H + 4}

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Translational thermal nonequilibrium in plasmas Thermalization

Electron momentum relation

The projection of the Boltzmann eq. at order ε−1 on the collisional invariants ˆ ψl

e, l ∈ {1, 2}, is trivial

Momentum is not included in the electron collisional invariants since

• Fe(φe), C

e

e = 0

At order ε−1, the zero-order momentum transfered from electrons to heavy particles reads

• j∈H
• J0

ej(f 0 e φe, ˆ

f 0

j ), C e

e = 1 Mh ∂xpe − neqe Mh E A 1storder electron momentum is also derived at order ε0

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Translational thermal nonequilibrium in plasmas Conservation equations

Zero-order drift-diffusion and Euler eqs.

The electrons diffuse in the vh reference frame ∂tρe + ∂x·(ρevh) = − 1

Mh ∂x·(ρeV e)

∂t(ρeee) + ∂x· (ρeeevh) + pe∂x·vh = − 1

Mh ∂x·qe + 1 Mh J e·E′ + ∆E 0 e

The heavy particles are coupled with the electrons through a pressure gradient and Lorentz force ∂tρi + ∂x·(ρivh) = 0, i ∈ H ∂t(ρhvh) + ∂x·(ρhvh⊗vh +

1 M2

h pI)

=

1 M2

h nqE + δb1I0∧B

∂t(ρheh) + ∂x·(ρhehvh) + ph∂x·vh = ∆E 0 with the zero-order energy exchange terms (Jeans’ relaxation) ∆E 0 + ∆E 0

e

= ∆E 0 =

3 2ne(Te − Th) 1 τ

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Translational thermal nonequilibrium in plasmas Conservation equations

1st order drift-diffusion and Navier-Stokes eqs.

1st and 2nd order transport fluxes for the electrons

∂tρe+∂x·(ρevh)

=

− 1 Mh ∂x·[ρe(V

e +εV2 e )]

∂t(ρeee)+∂x·(ρeeevh)+pe∂x·vh

=

− 1 Mh ∂x·(qe+εq2

e)+ 1

Mh (J

e+εJ2 e )·E′+δb0εMhJ e·vh∧B

+∆E 0

e +ε∆E 1 e

1st order transport fluxes for the heavy particles

∂tρi+∂x·(ρivh)

=

− ε Mh ∂x·(ρiV

i ),

i∈H ∂t(ρhvh)+∂x·(ρhvh⊗vh+ 1 M2

h

pI)

=

− ε M2

h

∂x·Π

h+ 1

M2

h

nqE+(δb0I0+δb1I)∧B ∂t(ρheh)+∂x·(ρhehvh)+ph∂x·vh

=

−εΠ

h:∂xvh− ε

Mh ∂x·qh+ ε Mh J

h·E′+∆E 0+ε∆E 1 h

with 1st order energy exchange terms ∆E 1

h + ∆E 1 e

= ∆E 1

h

=

• j∈H

njV

j ·F je

and average electron force acting on the heavy particles F

ie =

• Q(1)

ie (|C e|2) |C e|C e f 0 e φe dC e,

i ∈ H

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Transport phenomena: closure at the microscopic level

Macroscopic description of transport phenomena in a gas

⇒ conservation of mass momentum and energy

in a small element of volume of gas Transport fluxes

Diffusion of chemical species and heat fluxes arise through the interface Shear stresses applied on the surface of the volume

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

1st order electron perturbation function (b < 0 and b = 0)

Proposition (Degond, Lucquin 1996, Graille, M., Massot 2009)

The scalar function φe = −peφD

e

e ·de − φλ′

e

e ·∂x

• 1

Te

• is the solution to the

Boltzmann eq. at order ε−1 where the vectorial functions φD

e

e

and φλ′

e

e

are the solutions to the eqs. Fe(φµ

e ) = Ψµ e ,

with µ ∈ {De, λ′

e}

under the scalar constraints f 0

e φµ e , ˆ

ψl

e

e = 0, l ∈ {1, 2} The electron diffusion driving force is de = 1

pe (∂xpe − neqeE)

An electron bracket operator is introduced [ [ξe, ζe] ]e = f 0

e ξe, Fe(ζe)

e

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

1st order electron perturbation function (b < 0 and b = 0)

Proposition (Degond, Lucquin 1996, Graille, M., Massot 2009)

The electron diffusion velocity reads V

e = −Dede − θe∂xlnTe

the electron heat flux qe − ρeheV

e = −λ′ e∂xTe − peθede

and the electron viscous tensor vanishes, i.e., Πe = 0

Diffusion coefficient: De = 1

3 peTeMh[

[φD

e

e , φD

e

e ]

]e Thermal diffusion coefficient: θe = − 1

3 Mh[

[φD

e

e , φλ′

e

e ]

]e Partial thermal conductivity: λ′

e = 1 3T 2

e Mh[

[φλ′

e

e , φλ′

e

e ]

]e

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

2nd order electron perturbation function (b < 0 and b = 0)

The solution to the Boltzmann eq. at order ε0 is φ(2)

e

= −φηe

e :∂xvh − δb0peφD

e

e ·d2 e − pe

P

j∈H

φ

D

j

e ·d2 j − e

φ2

e

with the 2nd diffusion driving forces d2

e = −neqeM2 hvh∧B/pe and d2 i = −Vi,

i ∈ H

Proposition (Graille, M., Massot 2009)

The second-order electron diffusion velocity is V2

e = −δb0Ded2 e − j∈H

αejd2

j

The second-order electron heat flux q2

e − ρeheV2 e = −δb0peθed2 e − pe

• j∈H

χe

jd2 j

The average electron force acting on i heavy particles F

ie = − pe niMh αeide − pe niMh χe i ∂xlnTe

i ∈ H

With the αei coefficients and second-order electron thermal diffusion ratios

αei = 1

3peTeMh[

[φD

e

e , φD

i

e ]

]e χe

i = − 1 3Mh[

[φD

i

e , φλ′

e

e ]

]e, i ∈ H

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Kolesnikov effect

The second-order electron diffusion velocity and heat flux are also proportional to the heavy-particle diffusion velocities We refer to this coupling phenomenom as the Kolesnikov effect (1974)

This coupling is essential to derive a total energy eq. and entropy eq. that satisfy the laws of thermodynamics These 2nd order transport fluxes should not be confused with Burnett transport fluxes

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Kolesnikov effect

The heavy-particle diffusion velocities V

i = −

• j∈H

Dij ˆ dj − θh

i ∂xlnTh,

i ∈ H are proportional to

The diffusion driving forces ˆ di =

1 ph ∂xpi − niqi ph E − niMh ph F ie

The heavy-particle temperature gradient (Soret effect)

The average electron force F

ie contributes to the diffusion driving

force ˆ di

The average electron force acting on the heavy particles is expressed in terms of the electron driving force and temperature gradient F

ie = − pe niMh αeide − pe niMh χe i ∂xlnTe

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Kinetic data

⇒ Closure of the transport fluxes at a microscopic scale

The transport properties are expressed in terms of collision integrals The kinetic data are the potentials for the particle binary interactions “Boltzmann impression”, Losa, Luzern 2004

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 29 / 59

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Kinetic data

The bracket operators [ [·, ·] ]e are reduced in terms of collisions integrals Transport collision integrals are average cross-sections over velocity Closure of the conservation equations is realized at the microscopic level by determining from experimental measurements

Either the potentials of interaction between the gas particles Or the cross-sections [paper AIAA 2002-2226]

b χ

Dynamics of an elastic binary collision

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Charge-charge and electron-neutral interactions

Charge-charge interactions

Shielded Coulomb potential [Mason et al] and [Devoto]: ϕ(r) = ±ϕ0 d r exp

• − r

d

• Electron-neutral interactions

2500 5000 7500 10000 1250 Temperature [ K ] 5 10 15 20 25 30 35 40

2 (1,1)* [ 10

• 20m

2 ]

e-CO2 e-C e-CO e-O e-O2

Recommended momentum- transfer cross-sections:

[Itikawa] e − CO2, e − CO, e − O2 and e − O [Thomas and Nesbet] e − C

Ω(2,2)⋆ = Ω(1,1)⋆

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Conditions on the kinetic data

Well-posedness of the transport properties is established provided that some conditions are met by the kinetic data For instance, the electrical conductivity and thermal conductivity reads in the first and second Laguerre-Sonine approximations, respectively

σe(1) =

4 25 (xeqe)2 k2

BTe

1 Λ00

ee

λe(2) = x2

e

Λ11

ee

Proposition (M. and Degrez, 2004)

Let ¯ Q(1,1)

ie

, ¯ Q(1,2)

ie

, ¯ Q(1,3)

ie

, i ∈ H and ¯ Q(2,2)

ee

be positive coefficients such that 5 ¯ Q(1,2)

ie

− 4 ¯ Q(1,3)

ie

< 25 ¯ Q(1,1)

ie

/12, and assume that xi > 0, i ∈ S. Then the scalars Λ00

ee and Λ11 ee are positive

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Mutation++ library

MUTATION++: MUlticomponent Transport And Thermodynamic properties / chemistry for IONized gases written in C++

Mixture

Thermo- dynamics Thermo- dynamic Databases

Multi- Temperature Lookup Tables Rigid- Rotator & Harmonic- Oscillator NASA Polynomials

State Model

CR Multi-T

Transport Collision Database Algorithms Kinetics Reaction Mech- anisms Rate Laws Species Jacobian Finite-rate Chemistry

art @ www.annelindner.com

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Translational thermal nonequilibrium in plasmas Transport fluxes and coefficients

Electron transport coefficients

Electron conduction current density: J

e

= neqeV

e

= σeE + · · ·

2500 5000 7500 10000 12500 15000

T [K]

10 10

1

10

2

10

3

10

4

σe [S m

• 1]

ξ=1 ξ=2

Electrical conductivity of carbon dioxide at 1 atm

− − σe(1) Mutation , σe(2) Mutation, and × Andriatis and Sokolova

with σe= (neqe)2

pe De

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 34 / 59

SLIDE 35

Atomic ionization reactions

• 2. Atomic ionization reactions

[Graille, Magin, Massot, CTR SP 2008] [Magin, Graille, Massot, CTR ARB 2009] [Massot, Graille, Magin, RGD 2010] Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 35 / 59

SLIDE 36

Atomic ionization reactions

UTIAS shock-tube experiments, Glass and Liu (1978)

(Mach=15.9, p=5.14 Torr, T=293.6 K, α=0.14)

Mass density and electron number density [Kapper and Cambier, 2011]

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 36 / 59

SLIDE 37

Atomic ionization reactions

Boltzmann equation with reactive collisions

Assumptions

Plasma spatially uniform, at rest, no external forces Composed of electrons, neutral particles, and ions: S = {e, n, i} Ionization mechanism: reaction r

i

n + ˙ ı ⇋ i + e + ˙ ı, ˙ ı ∈ S Maxwellian regime for reactive collisions (chemistry characteristic times larger than the mean free times)

Boltzmann eq.1 : ∂t⋆f ⋆

i

=

j∈S

J⋆

ij

• f ⋆

i , f ⋆ j

• + C⋆

i (f ⋆),

i ∈ S Reactive collision operator for particle i: C⋆

i = Cr

e⋆

i

+ Cr

n⋆

i

+ Cr

i⋆

i

1Dimensional quantities are denoted by the superscript ⋆ Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 37 / 59

SLIDE 38

Atomic ionization reactions

Reactive collision operator

e.g., e-impact ionization reaction r

e

n + e ⇋ i + e + e For electrons Cr

e⋆

e (f ⋆) =

f ⋆

i f ⋆ e1f ⋆ e2

β⋆

i β⋆ e

β⋆

n

− f ⋆

n f ⋆ e

• Wiee

ne ⋆dc⋆ ndc⋆ i dc⋆ e1dc⋆ e2

− 2 f ⋆

i f ⋆ e f ⋆ e2

β⋆

i β⋆ e

β⋆

n

− f ⋆

n f ⋆ e1

• Wiee

ne ⋆ dc⋆ ndc⋆ i dc⋆ e1dc⋆ e2,

with the statistical weight β⋆

i = [hP/(aim⋆ i )]3, ae = 2,an = ai = 1

For ions Cr

e⋆

i (f ⋆)

= − f ⋆

i f ⋆ e2f ⋆ e3

β⋆

i β⋆ e

β⋆

n

− f ⋆

n f ⋆ e1

• Wiee

ne ⋆ dc⋆ ndc⋆ e1dc⋆ e2dc⋆ e3

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 38 / 59

SLIDE 39

Atomic ionization reactions

Dynamic of the reactive collisions

e-impact ionization n + ¯ e ⇋ i + ˆ e + ˜ e |cn|2 = |ci|2 + O(ε) |c¯

e|2 = |cˆ e|2 + |c˜ e|2 + 2∆E + O(ε), with the ionization energy ∆E = UF

e + miUF i − mnUF n

Heavy-particle impact ionization n +¯ i ⇋ i + ˆ e +˜ i, i ∈ H

1 2m˙ ı|gn˙ ı|2 − 2∆E = 1 2m˙ ı|g′ i˙ ı|2 + O(ε),

˙ ı ∈ H |g′

he|2 = O(ε)

⇒ the electron pulled from the neutral particle is cold

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 39 / 59

SLIDE 40

Atomic ionization reactions

Euler conservation equations (order ε−1)

Mass dtρe = ω0

e

dtρi = mi ω0

i ,

i ∈ H Energy dt(ρeeT

e )

= −∆E 0 − ∆E ωr

e0

e

dt(ρheT

h )

= ∆E 0 + ∆E ωr

n0

n

− ∆E ωr

i0

i

Chemical loss rate controlling energy [Panesi et. al, JTHT 23 (2009) 236] Standard derivation [Appleton & Bray] does not account for mass disparity

Using the property ωr0

e = ωr0 i = −ωr0 n , r ∈ R, the mixture mass and

energy are conserved, i.e., dtρ = 0, dt(ρeT + ρUF) = 0

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 40 / 59

SLIDE 41

Atomic ionization reactions

Two temperature Saha law

e-impact ionization n + ¯ e ⇋ i + ˆ e + ˜ e Keq

r

e (Te) =

mi mn 3/2 QT

e (Te) exp

• −∆E

Te

• Heavy-particle impact ionization

n +¯ i ⇋ i + ˆ e +˜ i Keq

r

i (Th, Te) =

mi mn 3/2 QT

e (Te) exp

• −∆E

Th

• ,

i ∈ H

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 41 / 59

SLIDE 42

Atomic ionization reactions

Law of mass action for plasmas

e-impact ionization n + ¯ e ⇋ i + ˆ e + ˜ e Kf

r

e = Kf

r

e(Te),

Kb

r

e = Kb

r

e(Te)

Heavy-particle impact ionization n +¯ i ⇋ i + ˆ e +˜ i Kf

r˙

ı = Kf

r˙

ı(Th),

Kb

r˙

ı = Keq

r

i (Th, Te)Kf

r˙

ı(Th),

i ∈ H

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 42 / 59

SLIDE 43

Atomic ionization reactions

Thermo-chemical dynamics and chemical quasi-equilibrium

The species Gibbs free energy is defined as ρigi = niTi ln

• ni

QT

i (Ti)

• + ρiUF

i ,

i ∈ S Modified Gibbs free energy for thermal non-equilibrium ρi ˜ gr

j

i = ρigi +

• Ti

Tr

j − 1

• ρiUF

i ,

i, j ∈ S ⇒ The 2nd law of thermodynamics is satisfied dt(ρs) = Υ

th + j∈S Υr

j

ch,

Υ

th ≥ 0,

Υr

j

ch ≥ 0, j ∈ S The full thermodynamic equilibrium state of the system under well-defined and natural constraints is studied by following Giovangigli and Massot (M3AS 1998) The system asymptotically converges toward a unique thermal and chemical equilibrium

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 43 / 59

SLIDE 44

Internal energy excitation in molecular gases

• 3. Internal energy excitation in molecular gases

[Magin, Graille, Massot, AIAA 2011] [Magin, Graille, Massot, RGD 2012] [Munaf´

• , Haack, Gamba, Magin, RGD 2012]

[Torres, Magin, RGD 2012] [Munaf´

• , Torres, Haack, Gamba, Magin AIAA 2013]

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 44 / 59

SLIDE 45

Internal energy excitation in molecular gases

Microscopic approach to derive macroscopic nonequilibrium models...

e.g. NASA ARC database for nitrogen chemistry:

9390 (v,J) rovibrational energy levels for N2 50 × 106 reaction mechanism for N2 + N system

N2(v, J) + N ↔ N + N + N N2(v, J) ↔ N + N N2(v, J) + N ↔ N2(v′, J′) + N

Papers AIAA 2008-1208, 2008-1209, 2009-1569,

2010-4517, RTO-VKI LS 2008 N3 Potential Energy Surface NASA Ames Research Center Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 45 / 59

SLIDE 46

Internal energy excitation in molecular gases

Detailled chemistry of N2

Normal shock: T1 = 300 K, p1 = 13 Pa, u1 = 10 km/s, xN1 ∼ 2.8%

Rovibrational N2 population at 7mm

0.02 0.03 0.04 0.05 0.06 0.07

T

• 1/3

10

• 9

10

• 8

10

• 7

10

• 6

10

• 5

τ p [atm-s]

τRot τVib τInt

M.&W. τVib

Energy relaxation times in 0D reactor

The assumption of equilibrium between the rotational and translational modes is questionable... At high temperature the three relaxation times converge [Panesi, Magin, Jaffe, Schwenke, 2013]

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 46 / 59

SLIDE 47

Internal energy excitation in molecular gases

Internal energy excitation in molecular gases

Wang-Chang-Uhlenbeck quasi-classical description

The gas is composed of identical particles with internal degrees of freedom The particles may have only certain discrete internal energy levels These levels are labelled with an index i, with the set of indices I Quantity E ⋆

i stands for the energy of level i ∈ I, and ai, its

degeneracya

aDimensional quantities are denoted by the superscript ⋆ Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 47 / 59

SLIDE 48

Internal energy excitation in molecular gases

Collision zoology according to Ferziger and Kaper

Elastic collisions (i, j) ⇋ (i, j), i, j ∈ I

⇒ Both kinetic and internal energies are conserved: E i′j′⋆

ij

= 0

Inelastic collisions (i, j) ⇋ (i′, j′), i, j, i′, j′ ∈ I, (i′, j′) = (i, j)

General case: E i′j′⋆

ij

= 0 Resonant collisions: E i′j′⋆

ij

= 0

e.g. exchange collision: (i, j) ⇋ (j, i), i, j ∈ I, i = j

Quasi-resonant collisions: E i′j′⋆

ij

∼ 0

Fast collisions associated with the energy level i ∈ I

Fi = {(j, i′, j′) ∈ I3 and |E i′j′⋆

ij

| ≤ εkBT 0} Elastic collisions & resonant and quasi-resonant inelastic collisions

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 48 / 59

SLIDE 49

Internal energy excitation in molecular gases Boltzmann equation

Assumptions

1

There are no external forces

2

The reactive collisions are not accounted for

3

Boltzmann collision operator for inert collisions: (i, j) ⇋ (i′, j′), i, j, i′, j′ ∈ I

(i, j) and (i′, j′) are ordered pairs of energy levels with the net internal energy E i′j′⋆

ij

= E ⋆

i′ + E ⋆ j′ − E ⋆ i − E ⋆ j 4

Two categories of collisions based on thermal energy threshold

Fast collisions: |E i′j′⋆

ij

| ≤ εkBT 0, with ref. differ. cross-section σ0 Slow collisions: |E i′j′⋆

ij

| > εkBT 0, with ref. differ. cross-section εσ0

5

The macroscopic time scale is t0 = τ 0/ε, with the kinetic time scale for fast collisions τ 0. The macroscopic length scale is L0 = v 0t0, where quantity v 0 is a reference hydrodynamic velocity

6

The pseudo Mach number, M = v 0/V 0, is of order one

The Knudsen number is

Kn = l0/L0 = V 0τ 0/(v0t0) = ε/M

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 49 / 59

SLIDE 50

Internal energy excitation in molecular gases Boltzmann equation

Boltzmann equation

The temporal evolution of f ⋆

i (t⋆, x⋆, c⋆ i ) is governed by

∂t⋆f ⋆

i

+ c⋆

i ·∂x⋆f ⋆ i

= J⋆

i (f ⋆),

i ∈ I The collision operator comprises 2 contributions J⋆

i (f ⋆) = JF⋆ i

(f ⋆) + JS⋆

i (f ⋆)

Fast collision operator: JF⋆

i

(f ⋆) =

(j,i′,j′)∈Fi Ji′j′⋆ ij

(f ⋆

i , f ⋆ j )

Slow collision operator: JS⋆

i

(f ⋆) =

(j,i′,j′)∈Si Ji′j′⋆ ij

(f ⋆

i , f ⋆ j )

with the partial collision operators Ji′j′⋆

ij

(f ⋆

i , f ⋆ j ) =

f ⋆

i′ f ⋆ j′ aiaj ai′aj′ − f ⋆ i f ⋆ j

• W i′j′⋆

ij

dc⋆

j dc⋆ i′dc⋆ j′

Property (collisional invariants)

The collision operator J⋆ is orthogonal to the space of collisional invariants, i.e., ψl⋆, J⋆ ⋆ = 0, for all l ∈ {1, . . . , 5}.

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 50 / 59

SLIDE 51

Internal energy excitation in molecular gases Boltzmann equation

Collisional invariants

Although the number of particles in each energy level is not conserved in a collision, the total number of particles (mass), momentum, and total energy are conserved    ψ1⋆

i

= m⋆ ψ1+ν⋆

i

= m⋆c⋆

iν,

ν ∈ {1, 2, 3} ψ5⋆

i

=

1 2m⋆c⋆ i ·c⋆ i + E ⋆ i

The individual contributions to the total energy, i.e., the kinetic energy 1

2m⋆c⋆ i ·c⋆ i and the internal energy E ⋆ i , are not conserved

through collisions The kernel of the fast collision operator JF⋆

i

(f ⋆) is also spanned by ψl⋆, l ∈ {1, . . . , 5}

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 51 / 59

SLIDE 52

Internal energy excitation in molecular gases Boltzmann equation

Flow macroscopic properties

The collisional invariants allow to introduce the flow macroscopic quantities mass, momentum, and total energy, as average microscopic quantities:    ρ⋆ =

• f ⋆, ψ1⋆

⋆ ρ⋆v⋆

ν

=

• f ⋆, ψ1+ν⋆

⋆, ν ∈ {1, 2, 3}

1 2ρ⋆|v⋆|2 + ρ⋆e⋆ + ρ⋆E⋆

=

• f ⋆, ψ5⋆

⋆ The flow total energy is the sum of the hydrodynamic kinetic energy, translational energy, and internal energy ⇒ How can we separate the contributions of the translational energy from the internal energy?

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 52 / 59

SLIDE 53

Internal energy excitation in molecular gases Boltzmann equation

Dimensional analysis

Reference dimensional quantities Number density n0 Thermal speed V 0 Fast differ. cross-section σ0 Fast kinetic time scale τ 0 Mean free path l0 Macroscopic time scale t0 Temperature T 0 Macroscopic length L0 Mass m0 Hydrodynamic velocity v0 Non dimensional Boltzmann eq. Di(f

i ) = 1

εJF

i (f) + JS i (f),

i ∈ I The multiscale analysis occurs at three levels:

In the righ-hand-side of the Boltzmann eq. In the energy collisional invariant ψ5

i for fast collisions

In the fast collision operator JF

i (f)

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 53 / 59

SLIDE 54

Internal energy excitation in molecular gases Boltzmann equation

Perturbed energy level lemma

The following lemma allows to split the internal energy of all the levels in perturbed elastic and inelastic contributions for the fast collisions

Lemma (Graille, M., Massot 2012)

For all energy level i ∈ I, let us consider the fast collisions (i, j) ⇋ (i′, j′), with (j, i′, j′) ∈ Fi = {(j, i′, j′) ∈ I3 and |E i′j′

ij | ≤ ε}. There is a perturbed

energy ˆ Ei ∈ R such that |Ei − ˆ Ei| ≤ Cε, where C is a constant, and such that for all the fast reactions, the net perturbed energy vanishes, i.e., ˆ E i′j′

ij

= 0. ⇒

This property is crucial to separate the energy collision invariant into fast collisional invariants and to expand JF

i (f) in ε

In turn, it allows for a generalization of the Chapman-Enskog method to gases with internal degrees of freedom in thermal nonequilibrium A particular example is the theory for the generalized Treanor distribution [Kustova and Nagnibeda 2009]

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 54 / 59

SLIDE 55

Internal energy excitation in molecular gases Boltzmann equation

Fast collision invariants

The fast scalar collisional invariants are introduced as          ˆ ψ1

i

= m ˆ ψ1+ν

i

= mciν, ν ∈ {1, 2, 3} ˆ ψ5

i

=

1 2mci·ci

ˆ ψ6

i

= ˆ Ei The fast collision operator JF

i = (JF i )i∈I obeys the following property

Property (Fast collision invariants)

The collision operator JF is orthogonal to the space of fast collisional invariants, i.e., ˆ ψl, JF = 0, for all l ∈ {1, . . . , 6}. Hydrodynamic kinetic energy and translational energy are obtained based on the microscopic kinetic energy: 1

2ρ|v|2 + ρe =

f, ˆ ψ5

• Macroscopic internal energy: ρ ˆ

E = f, ˆ ψ6

• Thierry MAGIN (VKI)

Multiscale asymptotic solutions 3 June 2013 55 / 59

SLIDE 56

Internal energy excitation in molecular gases Boltzmann equation

Relaxation towards equilibrium of a multi-energy level gas

Translational and internal degrees of freedom initially in equilibrium at their own temperature

ρ = 1kg/m3, T = 1000 K, Tint = 100 K 5 levels, Anderson cross-section model

Unbroken lines: Spectral Boltzmann Solver [Munafo et al. 2013], symbols: DSMC

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 56 / 59

SLIDE 57

Internal energy excitation in molecular gases Boltzmann equation

Flow across a normal shockwave for multi-energy level gas

Free stream conditions

ρ∞ = 10−4kg/m3, T∞ = 300 K, v∞ = 954 m/s 2 levels, Anderson cross-section model

Unbroken lines: Spectral Boltzmann Solver [Munafo et al. 2013], symbols: DSMC

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 57 / 59

SLIDE 58

Conclusion

Final thoughts

Plasmadynamical models based on a multiscale Chapman-Enskog perturbative method

Scaling derived from a dimensional analysis of the Boltzmann eq. Collisional invariants identified in the kernel of collision operators Macroscopic conservation eqs. follow from Fredholm’s alternative Laws of thermodynamics and law of mass action are satisfied Well-posedness of the transport properties is established, provided that some conditions on the kinetic data are met

Advantages compared to conventional models for atmospheric entry plasma flows

Mathematical structure of the conservation equations well identified Rigorous derivation of a set of macroscopic equations where hyperbolic and parabolic scalings are entangled [Bardos, Golse, Levermore 1991] The mathematical structure of the transport matrices is readily used to build transport algorithms (direct linear solver / convergent iterative Krylov projection methods) [Ern and Giovangigli 1994, Magin and Degrez 2004]

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 58 / 59

SLIDE 59

Conclusion

Future work

New exploratory field: rarefied gas effects

Riemann problem for ionized gases (non conservative products) (with

Marc Massot and Benjamin Graille)

Boltzmann moment systems with Grad closure for plasmas (with G´

erald Martins, Mike Kapper and Manuel Torrilhon)

Development of deterministic Boltzmann solver for atmospheric entry flows (with Alessandro Munaf`

• , Irene Gamba and Jeff Haack)

Physically / numerically consistent kinetic and continuum flow solvers

(with Alessandro Munaf`

• , Erik Torres, Phil Varghese, David Goldstein, Peter Clarke,

Irene Gamba and Jeff Haack)

⇒ Poster: “A Spectral-Lagrangian Boltzmann Solver for a Multi-Energy Level Gas”, Alessandro Munaf`

• Future work

Dissociation of molecular gases Radiation New application: meteoroids

Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 59 / 59